Read the requirements for each problem and plan your responses carefully. Ensure that you answer each of the required questions as concisely and as completely as possible and include supporting calculations where required.
1. A stock sells for $52 per share, and the 6-month European call on the stock with a strike price of $50 sells for $2.50. The stock is not expected to pay any dividends in next six months. The risk free interest rate is 4% per annum, continuously compounded. How can you get a free lunch from the market? Describe your transactions clearly.
2. An at-the-money three-month European call option on a non-dividend-paying stock has a market price of $1.27. The stock price is $20 and the risk-free interest rate is 5% per annum, with continuous compounding. Verify that the implied volatility is about 27%.
3. A European call option and put option on a stock both have a strike price of $20 and an expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19, and no dividend is expected in three months. How can you make a free lunch in these markets?
4. A non-dividend-paying stock sells for $42 per share. The continuously compounded risk-free interest rate is 6% per annum, and the volatility of the stock price is 30% per annum. Use the Black-Scholes-Merton model to determine the price of a 3-month European call on the stock with a strike price of $40.
5. A stock price is currently $60, with an annual volatility of 0.30. The risk-free rate is 4% per annum. Use the two-period binomial model to
a. calculate the price of a one-year European put option on the stock with a strike price of $60.
b. calculate the price of a one-year American put option on the stock with a strike price of $60.
6. Stock ABC sells for $64 and is not to pay any dividend in next year. Several 6-month European options on MFN are listed below with their market prices:
|
Option
name
|
Type
|
Strike price
|
Market
price
|
N(d1)
|
N(d2)
|
|
A
|
Call
|
$60
|
8.4
|
0.70
|
0.62
|
|
B
|
Call
|
$65
|
5.8
|
|
C
|
Call
|
$70
|
3.7
|
|
D
|
Put
|
$60
|
3.0
|
|
E
|
Put
|
$65
|
5.1
|
|
F
|
Put
|
$70
|
| |
|
|
|
|
|
|
|
Assume the continuously compounded interest rate is 6% per annum for all terms and the volatility of Stock ABC is 0.3.
a. What are the prices of option A and D according to the Black-Scholes-Merton model?
b. If the stock price changes to $64.5, while other variables stay the same, what would be your estimates of the market price of Option A?
c. If the stock price changes to $63.2, while other variables stay the same, what would be your estimates of the market price of Option D?
d. Assume that Option B has a delta of 0.56. The probability that the option will be exercised on maturity date is 0.48. Use the B-S-M model to determine if Option B is overpriced, fairly priced, or underpriced?
e. Suppose you short 100 Option B, how many shares do you need to hedge your position?
f. Suppose an investor expects the stock price to remain at about $64 and decides to execute a butterfly spread using options A, B, and C. What will be the profit if the stock price at expiration is $66.50?
g. Consider a long straddle constructed using the options with X=65. What are the two breakeven stock prices at expiration? What is the profit if the stock price at expiration is at $60?
h. Now suppose that ABC stock is expected to pay a continuous dividend yield of 2% per annum and option C has a fair price of 3.5. What should the price of Option F be?